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Why do Muslims only eat Halal meat and any other form of slaughtered meat is not permissible?

With respect to meat, the term Halal means "prepared appropriately for human consumption." The primary reason they do this is because they believe that God has instructed them to do so. There are other explanations from a scientific perspective (e.g. blood is probably unhealthy, pork is probably unhealthy), but even if Muslims didn't understand why it was this way, they'd do it out of obedience and trust in God.

Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah , and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows. That is gr***e disobedience. This day those who disbelieve h***e despaired of [defeating] your religion; so fear them not, but fear Me. This day I h***e perfected for you your religion and completed My f***or upon you and h***e approved for you Islam as religion. But whoever is forced by severe hunger with no inclination to sin - then indeed, Allah is Forgiving and Merciful.

So, if a food meets these criteria, then it is considered acceptable to eat or Halal. In practice this means:

The animal was slaughtered by a Muslim, or by a Jew or a Christian (since they share the same God) and they must slaughter it in the name of God.

The slaughter involves cutting the carotid artery of the animal and letting the blood drain. This is not in this verse, but other sayings make this clear.

Death from a blow to the head, or fallen animals, are not permitted.

The other sources also indicate that one must be merciful to the animal (e.g. not letting it see other animals be slaughtered, and making sure the blade is sharp so the slaughter is painless.